A ladder of length 13 m has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \mathrm{~m} / \mathrm{min}$ then the speed (in $\mathrm{m} / \mathrm{min}$ ) at which upper end falls when the bottom is 5 m away from the wall is

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is

If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$